Deterministic KPZ-type equations with nonlocal “gradient terms”

The main goal of this paper is to prove existence and non-existence results for deterministic Kardar–Parisi–Zhang type equations involving non-local “gradient terms”. More precisely, let Ω ⊂ RN, N≥ 2 , be a bounded domain with boundary ∂Ω of class C2. For s∈ (0 , 1) , we consider problems of the form {(-Δ)su=μ(x)|D(u)|q+λf(x),inΩ,u=0,inRN\Ω,(KPZ)where q> 1 and λ> 0 are real parameters, f belongs to a suitable Lebesgue space, μ∈ L∞(Ω) and D represents a nonlocal “gradient term”. Depending on the size of λ> 0 , we derive existence and non-existence results. In particular, we solve several open problems posed in [Abdellaoui in Nonlinearity 31(4): 1260-1298 (2018), Section 6] and [Abdellaoui in Proc Roy Soc Edinburgh Sect A 150(5): 2682-2718 (2020), Section 7]. © 2022, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.